H2 Consultation Exercise Note: Students should take 2.5 hour to complete the assignment. Answers are provided in [ ]. Topic: Differentiation Techniques 1. Differentiate with respect to x: a) sin −1 x b) ln k 2 + x 2 , where k is a constant c) ln 5 x ex 2  1   1+ x2  d) tan −1  e) 2. 2 ln x dy 2 x − 2 y − 1 = dx 2 x − 2 y + 1 i) For the curve ( x − y ) 2 = x + y , show that ii) Find the gradient of the curve at the points where it cuts the x-axis. Applications 1. The vertical cross-section of a water trough is in the shape of an equilateral triangle with one vertex pointing down. The trough is 15m long. When the water in the trough is 1m deep, its depth is increasing at a rate of 1 -1 ms . At what rate is water flowing 5 into the trough at that instant? 2. Two variables u and v are connected by the relation [ 2 3 m 3 / s] 1 1 1 + = , where f is a u v f constant. Given that u and v both vary with time, t , find an equation connecting du d v , , u and v . Given also that u is decreasing at a rate of 2cm per second and dt dt 1 that f =10cm, calculate the rate of increase of v when u = 50cm. [ cm/s] 8 3. A beam is to be cut from a cylindrical log so that its cross-section is a rectangle. The log has diameter d and the beam is to have breadth x and depth y . Given that the stiffness of such a beam is proportional to xy 3 , find, in terms of d , the values of x and y for the stiffest beam that can be cut from the log. [x = d 3 , y= d] 2 2 4. The parametric equations of a curve are x = ln(sin θ) , y = 3ln(cos θ) , 0 < θ < π . 2 Without the use of graphic calculator, find the equation of the tangent to the curve at the point where θ = π . 6 Will this tangent meet the curve again? Justify your answer. [ y = − x + ln 5. 3 3 , No] 16 A curve has parametric equations 1 x = 1− t 2 , y = , t ∈ , 0 < t ≤ 1 . t   1 t Find the equations of the tangent and normal to the curve at the point  1 − t 2 ,  .  3   2 , 2  cuts the y-axis at T while the normal to   The tangent to the curve at point P  the curve at the same point cuts the x-axis at N. Find the exact coordinates of T and N and deduce the area of triangle PTN. 1 t [ y− = 6. (a) 3   1 t 1− t 2  2 x − 1− t 2  , y − =−   x − 1− t  , 3  t 2   t 1− t  ( 0, −4 )  17 3  2 ,0  , 42.4 units ] 2   ,  The parametric equations of a curve are x = t ( t + 2 ) , y = 2 ( t + 1) . Find, in terms of p, the equation of the normal to the curve at P  p ( p + 2 ) , 2 ( p + 1)  . If this normal meets the x-axis at G and N is the foot of the perpendicular from P to the x-axis, find the length of NG. [ NG = 2 ] (b) The equation of a curve C is 2 + sin y = x 2 − xy . Find dy . dx Show that any tangent to C cannot be parallel to the x-axis. It is given instead that C has a tangent which is parallel to the y-axis. Show that the y-coordinate of the point of contact of the tangent with C must satisfy sin 2 y + sin y = y cos y − 1 .